Two liquids having vapour pressure `P_1^0 and P_2^0` in pure state in the ratio of `2:1` are mixed in the molar ratio `1:2` The ratio of their moles in the vapour state would be

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. `3:2`

Answer

To solve the problem of finding the ratio of moles of two liquids in the vapor phase after mixing, we will follow these steps:

### Step 1: Define the Vapour Pressures
Let the vapor pressures of the two liquids in their pure states be:
- \( P_1^0 = 2P \) (for liquid 1)
- \( P_2^0 = P \) (for liquid 2)

### Step 2: Define the Molar Ratios
The liquids are mixed in a molar ratio of 1:2. Therefore, if we take:
- Moles of liquid 1 = \( n \)
- Moles of liquid 2 = \( 2n \)

### Step 3: Calculate the Total Moles
The total number of moles in the mixture is:
\[ n_{\text{total}} = n + 2n = 3n \]

### Step 4: Calculate Mole Fractions in the Liquid Phase
The mole fractions of the two components in the liquid phase are:
- Mole fraction of liquid 1, \( x_1 \):
\[ x_1 = \frac{n}{n_{\text{total}}} = \frac{n}{3n} = \frac{1}{3} \]
- Mole fraction of liquid 2, \( x_2 \):
\[ x_2 = \frac{2n}{n_{\text{total}}} = \frac{2n}{3n} = \frac{2}{3} \]

### Step 5: Apply Raoult's Law
According to Raoult's Law, the partial pressures of the components in the vapor phase are given by:
- Partial pressure of liquid 1, \( P_1 \):
\[ P_1 = x_1 \cdot P_1^0 = \frac{1}{3} \cdot 2P = \frac{2P}{3} \]
- Partial pressure of liquid 2, \( P_2 \):
\[ P_2 = x_2 \cdot P_2^0 = \frac{2}{3} \cdot P = \frac{2P}{3} \]

### Step 6: Calculate Total Pressure
The total pressure \( P_t \) in the vapor phase is the sum of the partial pressures:
\[ P_t = P_1 + P_2 = \frac{2P}{3} + \frac{2P}{3} = \frac{4P}{3} \]

### Step 7: Calculate Mole Fractions in the Vapor Phase
Using the partial pressures, we can find the mole fractions in the vapor phase:
- Mole fraction of vapor 1, \( y_1 \):
\[ y_1 = \frac{P_1}{P_t} = \frac{\frac{2P}{3}}{\frac{4P}{3}} = \frac{2}{4} = \frac{1}{2} \]
- Mole fraction of vapor 2, \( y_2 \):
\[ y_2 = \frac{P_2}{P_t} = \frac{\frac{2P}{3}}{\frac{4P}{3}} = \frac{2}{4} = \frac{1}{2} \]

### Step 8: Find the Ratio of Moles in the Vapor Phase
The ratio of moles of the two components in the vapor phase is given by the ratio of their mole fractions:
\[ \text{Ratio of moles in vapor phase} = \frac{y_1}{y_2} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1:1 \]

### Final Answer
The ratio of their moles in the vapor state is **1:1**.

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